Simplify √12 Easy

The question

Question: simplify √12.

1 Find the biggest square number

Square numbers are a number times itself: 2 × 2 = 4, 3 × 3 = 9, and so on.

We list them up to half of 12.

A bigger one can't go in.

We try the biggest first.

Does 4 go into 12?

Yes: 12 divided by 4 is 3.

2 Split the root

So 12 is 4 times 3.

We write it under the root.

Rule: a root over a times can be split into two roots.

So √(4 × 3) is √4 times √3.

√4 is 2, because 2 times 2 is 4.

We write 2 in place of √4.

We leave out the times sign: 2√3 means 2 times √3.

3 has no square number in it, so we are done.

So √12 = 2√3.

3 Check

Let's check: we square the answer.

We must get 12 back.

2 × 2, and √3 × √3 = 3.

We get 12.

So 2√3 is right!

Let's look back.

First: find the biggest square number.

Then: split the root.

Last: check.

▶ Watch it solved

Simplify √72 Medium

The question

Question: simplify √72.

1 Find the biggest square number

Square numbers are a number times itself: 2 × 2 = 4, 3 × 3 = 9, and so on.

We list them up to half of 72.

A bigger one can't go in.

We try the biggest first.

Does 36 go into 72?

Yes: 72 divided by 36 is 2.

2 Split the root

So 72 is 36 times 2.

We write it under the root.

Rule: a root over a times can be split into two roots.

So √(36 × 2) is √36 times √2.

√36 is 6, because 6 times 6 is 36.

We write 6 in place of √36.

We leave out the times sign: 6√2 means 6 times √2.

2 has no square number in it, so we are done.

So √72 = 6√2.

3 Check

Let's check: we square the answer.

We must get 72 back.

6 × 6, and √2 × √2 = 2.

We get 72.

So 6√2 is right!

Let's look back.

First: find the biggest square number.

Then: split the root.

Last: check.

▶ Watch it solved

Simplify √18 + √32 - √2 Harder

The question

Question: simplify √18 + √32 − √2.

1 Find a square number in each

Square numbers are a number times itself: 2 × 2 = 4, 3 × 3 = 9, and so on.

We list them up to half of 32.

A bigger one can't go in.

First 18.

We try the biggest first.

Does 9 go into 18?

Yes: 18 divided by 9 is 2.

Now 32.

Does 16 go into 32?

Yes: 32 divided by 16 is 2.

2 Split each root

We write each number as the square times what is left.

Rule: a root over a times can be split into two roots.

So each root becomes two roots, times.

√9 is 3, because 3 times 3 is 9.

√16 is 4, because 4 times 4 is 16.

We write the numbers in place of the square roots.

We leave out the times sign: 3√2 means 3 times √2.

Careful!

We can't add the numbers under the roots.

√9 + √16 is 3 + 4 = 7, not √25 = 5.

3 Put like roots together

They all have √2.

They are like terms, like 3 and 4 and 1 of the same thing.

3 plus 4 minus 1 is 6.

So √18 + √32 − √2 = 6√2.

4 Check

Let's check with a calculator.

The start is about 8.49.

The answer is about 8.49 too.

So it is right!

Let's look back.

First: find a square number in each.

Then: split each root.

Then: put like roots together.

Last: check.

▶ Watch it solved